Integrand size = 24, antiderivative size = 144 \[ \int x^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=-\frac {a^3 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^4}+\frac {3 a^2 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^4}-\frac {3 a (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^4}+\frac {(a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^4} \]
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Time = 0.03 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {660, 45} \[ \int x^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^4}-\frac {3 a \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^4}+\frac {3 a^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^4}-\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^4} \]
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Rule 45
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^3 \left (a b+b^2 x\right )^5 \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {a^3 \left (a b+b^2 x\right )^5}{b^3}+\frac {3 a^2 \left (a b+b^2 x\right )^6}{b^4}-\frac {3 a \left (a b+b^2 x\right )^7}{b^5}+\frac {\left (a b+b^2 x\right )^8}{b^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = -\frac {a^3 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^4}+\frac {3 a^2 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^4}-\frac {3 a (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^4}+\frac {(a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^4} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.53 \[ \int x^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x^4 \sqrt {(a+b x)^2} \left (126 a^5+504 a^4 b x+840 a^3 b^2 x^2+720 a^2 b^3 x^3+315 a b^4 x^4+56 b^5 x^5\right )}{504 (a+b x)} \]
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Time = 2.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(\frac {x^{4} \left (56 b^{5} x^{5}+315 a \,b^{4} x^{4}+720 a^{2} b^{3} x^{3}+840 a^{3} b^{2} x^{2}+504 a^{4} b x +126 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 \left (b x +a \right )^{5}}\) | \(74\) |
default | \(\frac {x^{4} \left (56 b^{5} x^{5}+315 a \,b^{4} x^{4}+720 a^{2} b^{3} x^{3}+840 a^{3} b^{2} x^{2}+504 a^{4} b x +126 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 \left (b x +a \right )^{5}}\) | \(74\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{5} x^{9}}{9 b x +9 a}+\frac {5 \sqrt {\left (b x +a \right )^{2}}\, a \,b^{4} x^{8}}{8 \left (b x +a \right )}+\frac {10 \sqrt {\left (b x +a \right )^{2}}\, a^{2} b^{3} x^{7}}{7 \left (b x +a \right )}+\frac {5 \sqrt {\left (b x +a \right )^{2}}\, a^{3} b^{2} x^{6}}{3 \left (b x +a \right )}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{4} b \,x^{5}}{b x +a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{5} x^{4}}{4 b x +4 a}\) | \(153\) |
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Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.39 \[ \int x^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{9} \, b^{5} x^{9} + \frac {5}{8} \, a b^{4} x^{8} + \frac {10}{7} \, a^{2} b^{3} x^{7} + \frac {5}{3} \, a^{3} b^{2} x^{6} + a^{4} b x^{5} + \frac {1}{4} \, a^{5} x^{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (99) = 198\).
Time = 0.71 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.47 \[ \int x^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (- \frac {a^{8}}{504 b^{4}} + \frac {a^{7} x}{504 b^{3}} - \frac {a^{6} x^{2}}{504 b^{2}} + \frac {a^{5} x^{3}}{504 b} + \frac {125 a^{4} x^{4}}{504} + \frac {379 a^{3} b x^{5}}{504} + \frac {461 a^{2} b^{2} x^{6}}{504} + \frac {37 a b^{3} x^{7}}{72} + \frac {b^{4} x^{8}}{9}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {- \frac {a^{6} \left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7} + \frac {a^{4} \left (a^{2} + 2 a b x\right )^{\frac {9}{2}}}{3} - \frac {3 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {11}{2}}}{11} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {13}{2}}}{13}}{8 a^{4} b^{4}} & \text {for}\: a b \neq 0 \\\frac {x^{4} \left (a^{2}\right )^{\frac {5}{2}}}{4} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.91 \[ \int x^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=-\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} x}{6 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} x^{2}}{9 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4}}{6 \, b^{4}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a x}{72 \, b^{3}} + \frac {83 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2}}{504 \, b^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.74 \[ \int x^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{9} \, b^{5} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{8} \, a b^{4} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, a^{2} b^{3} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + a^{4} b x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, a^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) - \frac {a^{9} \mathrm {sgn}\left (b x + a\right )}{504 \, b^{4}} \]
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Timed out. \[ \int x^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int x^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]
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